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Properties

Label 2-75-15.2-c9-0-1
Degree $2$
Conductor $75$
Sign $0.00603 + 0.999i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−27.2 + 27.2i)2-s + (111. + 84.7i)3-s − 974. i·4-s + (−5.35e3 + 739. i)6-s + (−4.32e3 − 4.32e3i)7-s + (1.26e4 + 1.26e4i)8-s + (5.33e3 + 1.89e4i)9-s + 9.19e4i·11-s + (8.25e4 − 1.08e5i)12-s + (−9.65e3 + 9.65e3i)13-s + 2.35e5·14-s − 1.88e5·16-s + (2.37e5 − 2.37e5i)17-s + (−6.61e5 − 3.71e5i)18-s + 8.14e5i·19-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)2-s + (0.797 + 0.603i)3-s − 1.90i·4-s + (−1.68 + 0.232i)6-s + (−0.681 − 0.681i)7-s + (1.08 + 1.08i)8-s + (0.270 + 0.962i)9-s + 1.89i·11-s + (1.14 − 1.51i)12-s + (−0.0937 + 0.0937i)13-s + 1.64·14-s − 0.718·16-s + (0.688 − 0.688i)17-s + (−1.48 − 0.833i)18-s + 1.43i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00603 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.00603 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.00603 + 0.999i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.00603 + 0.999i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.268889 - 0.267272i\)
\(L(\frac12)\) \(\approx\) \(0.268889 - 0.267272i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-111. - 84.7i)T \)
5 \( 1 \)
good2 \( 1 + (27.2 - 27.2i)T - 512iT^{2} \)
7 \( 1 + (4.32e3 + 4.32e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 9.19e4iT - 2.35e9T^{2} \)
13 \( 1 + (9.65e3 - 9.65e3i)T - 1.06e10iT^{2} \)
17 \( 1 + (-2.37e5 + 2.37e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 8.14e5iT - 3.22e11T^{2} \)
23 \( 1 + (-6.20e5 - 6.20e5i)T + 1.80e12iT^{2} \)
29 \( 1 + 6.40e6T + 1.45e13T^{2} \)
31 \( 1 - 3.64e5T + 2.64e13T^{2} \)
37 \( 1 + (1.23e7 + 1.23e7i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.60e7iT - 3.27e14T^{2} \)
43 \( 1 + (-1.16e7 + 1.16e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.50e7 + 1.50e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (9.74e6 + 9.74e6i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.46e7T + 8.66e15T^{2} \)
61 \( 1 + 1.55e8T + 1.16e16T^{2} \)
67 \( 1 + (7.46e7 + 7.46e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 9.62e7iT - 4.58e16T^{2} \)
73 \( 1 + (2.25e8 - 2.25e8i)T - 5.88e16iT^{2} \)
79 \( 1 - 3.61e8iT - 1.19e17T^{2} \)
83 \( 1 + (5.68e7 + 5.68e7i)T + 1.86e17iT^{2} \)
89 \( 1 - 7.68e8T + 3.50e17T^{2} \)
97 \( 1 + (7.95e8 + 7.95e8i)T + 7.60e17iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.03000598220747850045126874537, −12.55277194551493147652912182084, −10.41134691417424954241653935516, −9.799936576625281269203779809894, −9.075656676172393273934041615558, −7.53859130056305756641561127595, −7.21702543613551949029660694851, −5.40624677821109256772190786904, −3.80907278984307328862171644347, −1.74285325744297718652599600263, 0.15484864237567553184727311668, 1.27802847369979635461556478617, 2.76095910701481164468348853483, 3.34261366261019664708724493782, 6.13905601652417973702721913473, 7.75908635369962190884369435384, 8.799227565059968549357375751007, 9.272443921953424192255269159499, 10.68138236818777028573113004642, 11.69851902357535179072836840998

Graph of the $Z$-function along the critical line